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arXiv:cond-mat/0703391 · Journal of the Physical Society of Japan

Effective Field Theory of Triangular-Lattice Three-Spin Interaction Model

Hiromi Otsuka

Published Jul 10, 2007• 4 citations
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Abstract

We discuss an effective field theory of a triangular-lattice three-spin interaction model defined by the ${\mathbb Z}_p$ variables. Based on the symmetry properties and the ideal-state graph concept, we show that the vector dual sine-Gordon model describes the long-distance properties for $p\ge5$; we then compare its predictions with the previous argument. To provide the evidences, we numerically analyze the eigenvalue structure of the transfer matrix for $p=6$, and we check the criticality with the central charge $c=2$ of the intermediate phase and the quantization condition of the vector charges.

DOI
10.1143/jpsj.76.073002

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We discuss an effective field theory of a triangular-lattice three-spin interaction model defined by the Zp{\mathbb Z}_pZp​ variables. Based on the symmetry properties and the ideal-state graph concept, we show that the vector dual sine-Gordon model describes the long-distance properties for p≥5p\ge5p≥5; we then compare its predictions with the previous argument. To provide the evidences, we numerically analyze the eigenvalue structure of the transfer matrix for p=6p=6p=6, and we check the criticality with the central charge c=2c=2c=2 of the intermediate phase and the quantization condition of the vector charges.
We provide an effective description of a triangular-lattice model with a three-spin interaction. On the basis of a symmetry consideration, we argue that a vector dual sine-Gordon field theory is relevant, and provides us a framework to analyze long-distance behaviors. In particular, we can characterize a criticality of an intermediate phase between the low- and high-temperature phases. Also, in comparison with a model on the Union Jack lattice, we explain that the sublattice symmetry of the triangular lattice plays a crucial role to exhibit a novel phase transition. Theoretical predictions obtained are then confirmed by numerical calculations.
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We study a Berezinskii-Kosterlitz-Thouless-like phase transition in a triangular-lattice three-spin interaction model. The vector sine-Gordon field theory describes defects involved, and enables us to perform the renormalization-group analysis of the transition. Based on it, we discuss properties of the spin-spin correlation function, and also analyze the helicity modulus for the present model. To provide the data to support our analysis, we perform the Monte-Carlo simulations to calculate them. Further, we exhibit snapshots of some typical defect conflgurations, which are helpful to illustrate the defect-mediated phase transition.
The triangular lattice antiferromagnet with S=1/2S=1/2S=1/2 spins and nearest-neighbor interactions is known to have long-range antiferromagnetic order, with nearest-neighbor spins at an angle of $120{}^{\ensuremath{\circ}}$. Numerical studies of quantum phases proximate to this state have been limited to small systems because of the sign problem in Monte Carlo simulations in imaginary time. We propose an effective lattice model for quantum fluctuations of the antiferromagnetic order and a sign-problem-free Monte Carlo algorithm enabling studies in large systems sizes. The model is a Z2{\mathbb{Z}}_{2}Z2​ gauge theory coupled to gauge-charged scalars which have a relativistic dispersion in the continuum limit. Crucially, the gauge theory is odd, i.e., there is a static, background Z2{\mathbb{Z}}_{2}Z2​ gauge charge on each site, accounting for the Berry phases of the half-odd-integer spins on each site. We present results of simulations on lattices of sizes up to $36\ifmmode\times\else\texttimes\fi{}36\ifmmode\times\else\texttimes\fi{}36$. Along with the antiferromagnetically ordered phase, our phase diagram has a valence bond solid state with a $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ unit cell and a gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquid. Deconfined critical points or phases in intermediate regions are not ruled out by our present simulations.
Transfer matrix methods are used to locate accurate phase boundary of the triangular lattice antiferromagnetic Ising model in magnetic field. Universal quantities such as the central charge and the first few scaling dimensions are obtained along the phase boundary except near the zero field point where the crossover effect degrades convergence. Numerical results are fully consistent with the operator content of the 3-state Potts model indicating that whole phase boundary belongs to the 3-state Potts universality class.